Transmissive Dynamics Of A SIR Epidemic Model With Criss-Cross Mechanism And Non-diffusive Susceptible Population

In this work we study the traveling waves for a SIR epidemic model with criss-cross mechanism and non-diffusive susceptible population.In addition,the asymptotic behaviour of this epidemic reaction-diffusion system is also a major concern under the condition of a localized spatial perturbation.Firstly,we introduce the research background and the proceeding of the epidemic model with criss-cross mechanism or non-diffusive susceptible population.Moreover,we elaborate on the issues to be studied in this paper.Through proper transformation,we discover that the model is a cooperative system.Secondly,when the basic reproduction number R0>1,we prove that there is a unique positive balance point in the cooperative system by using the phase-plane analysis method,and stability of the equilibrium point is obtained by analyzing the distribution of eigenvalues.Then the vector value upper and lower solutions of the system are constructed based on the linearization method,which proves the existence of nontrivial traveling waves and the minimal wave speed c*and the nonexistence of traveling waves for 0<c<c*.When R0<1,we show that the system admits no nontrivial traveling waves.Lastly,we consider the asymptotic spreading speed and the uniform persistence of the disease.For a local spatial perturbation cooperative system,we use the monotonic iteration method and strong maximum principle to prove the existence and uniqueness of the positive equilibrium.Besides,we prove that the solutions of such a system have an asymptotic speed of spread in large time via the comparison principle and auxiliary initial value problems.At length,the uniform persistence of the disease is obtained.